The velocity anisotropy - density slope relation

One can solve the Jeans equation analytically for equilibrated dark matter structures, once given two pieces of input from numerical simulations. These inputs are 1) a connection between phase-space density and radius, and 2) a connection between velocity anisotropy and density slope, the α − β relation. The first (phase-space density v.s. radius) has already been analysed through several different simulations, however the second (α− β relation) has not been quantified yet. We perform a large set of numerical experiments in order to quantify the slope and zero-point of the α− β relation. We find strong indication that the relation is indeed an attractor. When combined with the assumption of phase-space being a power-law in radius, this allows us to conclude that equilibrated dark matter structures indeed have zero central velocity anisotropy β0 ≈ 0, central density slope of α0 ≈ −0.8, and outer anisotropy of β∞ ≈ 0.5.